In my case (Moto G3, Android 6.0, Airdroid 4.0.0.4) I never got the External SD dialog, but still now an External HD that is visible on phone and web browser. From the web browser I can edit filenames in the phone directory, so have write permission, but the Cut/Paste options are greyed out, and more importantly the Download button is too, so I can't download files to my desktop.

It seems that Airdroid 1) does not realise the directory is r/w, 2) has disabled Download, which is not dependent on permissions anyway